June 2019 - mathfun@mailman.xmission.com

[math-fun] Rook 3×4 and 4×4 graphs are graceful
by Ed Pegg Jr 23 Jun '19

23 Jun '19

Both found by nickgard Rook move differences give all values 1 to 30. 0 1 9 30 16 29 2 19 22 3 27 7 Rook move differences give all values 1 to 48. 0 1 23 47 19 44 9 2 37 42 3 11 48 4 36 32 --Ed Pegg Jr

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[math-fun] Did any of you integrate √tan in freshman calculus?
by Bill Gosper 22 Jun '19

22 Jun '19

In[38]:= Integrate[√Tan[x], x] Out[38]= 2/3 Hypergeometric2F1[3/4, 1, 7/4, -Tan[x]^2] Tan[x]^(3/2) In[39]:= FunctionExpand@% Out[39]= 2/3 Tan[x]^( 3/2) (-((3 ArcTan[(-Tan[x]^2)^(1/4)])/(2 (-Tan[x]^2)^(3/4))) + ( 3 ArcTanh[(-Tan[x]^2)^(1/4)])/(2 (-Tan[x]^2)^(3/4))) I'm pretty sure this would've left me some PTSD. And again later with Macsyma's Risch implementation. I suspect a conspiracy to hide this from emotionally vulnerable undergrads. —rwg

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[math-fun] barriers
by Cris Moore 22 Jun '19

22 Jun '19

Here’s a history/philosophy question. We are often told that Gödel’s incompleteness theorems were a defeat for Hilbert’s program — that there was no method to find all the truths about mathematics, or even all the theorems. But in the modern age, we seem to view this as liberating — as a source of endless creativity, rather than a defeat — since it means mathematics will never be complete, and there will always be room for new proof techniques and new forms of reasoning. When did this shift first occur? Who first said “You mean no axiomatic system can prove everything, and no algorithm can tell us what it can prove? That’s great!” Any help tracking down early quotes along these lines would be greatly appreciated, for a piece that a friend of mine and I are writing. Best! - Cris p.s. I guess the same could be said about P vs. NP — who first said “some search problems are [probably] really hard, including telling whether short proofs exist? that’s great!"

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Re: [math-fun] Is it me, or is math.stackexchange.com controlled by morons?
by Dan Asimov 21 Jun '19

21 Jun '19

The actual question Bill asked there was how to figure the volume of an ellipsoid from its 3 principal circumferences (call them L_ab, L_bc, L_ca). That's a quite interesting question, because arclength of an ellipse is notoriously not in some nice closed form with elementary functions, yet there still might be some simple formula of form F(L_ab, L_bc, L_ca) = (4π/3)abc. where F should be symmetric in its 3 variables. Hmm, since the L's are 1-dimensional and volume is 3 dimensional, maybe just (*) K * L_ab * L_bc * L_ca = (4π/3)abc. Wouldn't that be nice! If so, then for the unit sphere we get K * (2π)^3 = 4π/3, or K = 1/(6 π^2) — IF (*) is correct in general. Now I'd like to test (*) on a noncircular ellipse whose exact semi-axes and circumference are known in simple terms. But maybe other people know better how to find such examples. (Needless to say, refuting (*) would require only a good numerical value.) —Dan Bill Gosper wrote: ----- https://math.stackexchange.com/questions/3268714/what-are-the-semiaxes-a-b-… -----

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[math-fun] Grooks
by James Propp 21 Jun '19

21 Jun '19

How is the work "grook" (coined by Piet Hein) supposed to be pronounced? According to https://en.wikipedia.org/wiki/Grook, it's spelled "gruk" in Danish, and http://www.pronunciationguide.info/Danish.html says that a Danish "u" is pronounced like the "u" in the American word "put", so I'm guessing that "grook" rhymes with "book" and not "spook". Am I right? Thanks, Jim Propp

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Re: [math-fun] Name that construction set
by Dan Asimov 21 Jun '19

21 Jun '19

I do recall some kind of construction kit that included slotted disks, but I think this had other pieces as well. As for Deely Bobbers, weren't they kid's beanies with two springs sprouting from the hat with little plastic things at the ends ... that bobbed? —Dan ----- It was definitely Deelie Bobbers! Thanks. (I misremembered the number of slits; it's eight rather than four. Also, I misremembered the name as "Deely Boppers", which is something else entirely.) -----

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[math-fun] Name that construction set
by James Propp 21 Jun '19

21 Jun '19

I remember from my childhood a construction set you could buy whose components were small plastic disks that had four radial slots in them, spaced at 90° angles. Each disk could have up to four other disks fitted into its slots. Does anyone recognize the construction set in question from this description of it? Thanks, Jim Propp

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Re: [math-fun] optimal advance tournament scheduling
by Keith F. Lynch 20 Jun '19

20 Jun '19

Tomas Rokicki <rokicki(a)gmail.com> wrote: > To generate inequivalent graphs, use either genreg or nauty/geng; > either should work fine (although I find genreg to be faster, even > though nauty is better known). Thanks, but I was looking for an algorithm I can include in my own program. I would be curious how many unlabeled (i.e. inequivalent) 8-vertex 16-edge graphs those find, however. Thanks. I did notice an interesting coincidence. The adjacency matrix (which is how I represent these graphs) is 8x8, the size of a chessboard. And to represent a 16-game tournament, I need 32 markers, the same number as there are chess pieces. Also, relabeling a graph, which is done by permuting the rows and columns in the same way, preserves the number of pieces on the white squares (hence also the number of pieces on the dark squares). (I'm not certain that those numbers aren't always equal to each other, however.) The adjacency matrix of Dan's graph of a cube with its body diagonals looks interesting: 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 It's unchanged under permuting the rows and columns in the same way. Is the same true of any other 8x8 symmetrical matrix with half the entries being 0 and the other half 1? (Other than the complement of this matrix, which is uninteresting for our purposes for obvious reasons.) And the pattern of 0s and 1s read left to right top to bottom I instantly recognized as the first 64 terms of the Thue-Morse sequence.

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Re: [math-fun] optimal advance tournament scheduling
by Dan Asimov 20 Jun '19

20 Jun '19

If for example A beats B and B beats C, then we can at least guess that A can beat C. So, even if A doesn't play C in this case, there seems to be at least some information about A vs. C. —Dan Michael Collins wrote: ----- Some quick observations: given two players who are adjacent in the ordering, we cannot tell who is stronger if they do not actually play one another; this means that a tournament schedule without a Hamiltonian path can never give enough information to reconstruct the entire ordering. It also means that an incomplete tournament schedule can never guarantee reconstruction of the full ordering. ... -----

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[math-fun] Fwd: Re: Dots and Triangles
by Bill Gosper 20 Jun '19

20 Jun '19

-------- Original Message -------- Subject: Re: [math-fun] Dots and Triangles Date: 2019-06-18 20:37 From: Mike Beeler <mikebeeler2(a)gmail.com> To: math-fun <math-fun(a)mailman.xmission.com> Reply-To: math-fun <math-fun(a)mailman.xmission.com> Very nice game! After losing many, I won twice, but can’t repeat it. It seems the program does not go for a maximum score, only for a (random?) win, thus making it hard to learn its strategy. — Mike Young Rohan remarks: Rohan Ridenour Tue, Jun 18, 8:30 PM (20 hours ago) There’s a very easy way to beat the AI. StringReverse@".sevom s'IA eht ypoc dna ,eno no ssap ,sbat owt nepO" It's hard to believe that such an elegant game wasn't invented sooner. What a shame not to have E. Berlekamp's insights into it. —rwg > On Jun 18, 2019, at 5:14 PM, Tomas Rokicki <rokicki(a)gmail.com> wrote: > > Dots and Boxes is a nice simple children's game, right? > Elwyn Berlekamp (the "wyn" in Winning Ways) showed us > the surprising depths to this game. > > Here's a tiny variant on a tiny board; just nine triangles. > You get to choose to go first or second. Can you beat it? > > I'm finding it surprisingly difficult. > > https://tomas.rokicki.com/dottri.html > > This should work on your mobile phone too. > > -tom > > -- > -- http://cube20.org/ -- http://golly.sf.net/ -- > _______________________________________________

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